Navigating Concave Regions in Continuous Facility

Location ProblemsRuilin Ouyang1, Michael R. Beacher2, Dinghao Ma1, and Md. Noor-E-Alam1,*

1Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, 02115, USA2Department of Industrial and Systems Engineering, Rutgers University, New Brunswick, NJ, 08854, USA*corresponding author email: mnalam@neu.edu

ABSTRACT

In prior literature regarding facility location problems, there has been little explicit acknowledgement of

problems arising from concave (non-convex) regions. This issue extends to computational geometry as

a whole, as there is a distinct deficiency in existing center finding techniques amidst work on forbidden

regions. In this paper, we present a novel method for finding representative regional center-points,

referred to as "Concave Interior Centers", to approximate inter-regional distances for solving optimal

facility location problems. The validity of the proposal as a means for solving these placements is

discussed on three "special cases", and on a humanitarian focused real-world application. We compare

the performance of the Concave Interior Center to the results from using Geometric Centers. We discuss

the implications of using externally located representative centers in facility location problems. The

context of the application involves maximizing the potential services available to homeless persons in

Fairfield County, Connecticut under a given budget through the construction of limited capacity night-by-

night shelters and optionally included food pantries. Our computational results show the efficiency of the

proposed techniques in solving a critical societal problem.

Keywords: Centers, Concave Regions, Distance measurement, Facility location problems, Geometric

distance, Continuous facility location, Homeless shelters.

1 IntroductionFinding regional centers and inter-regional distances is a critical component in solving many optimization

problems, specifically facility location problems. The point most often considered representative for a

region has been its Geometric Center:

( limn→∞

∑nk=1 xk

n, lim

n→∞

∑nk=1 yk

n)

The glaring issue for using this point is that it may fall exterior to the region it intends to define. While

there has been much work in dealing with forbidden regions, most has only considered convex regions,

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and very little has focused on finding points to represent regions when running models. This remark is

expounded upon in greater detail in the following section, and we do note that the arising issues have been

previously noted, though rarely with context1, 2. The existing workarounds proposed have typically been

for very specific case studies, and otherwise require problem relaxation3–7. We define a concave region as

any region containing two points between which a connecting straight line would fall at least partially

exterior to the region itself. This is to say, in other words, a non-convex region. Let us imagine a concave

region whose Geometric Center falls externally. A simple example of this occurrence is seen in Figure 1.

Figure 1. Map and outline of the primary districts of Shanghai (adopted from Google Maps).

The above figure shows Shanghai city map with its outline represented as two concave regions. Clearly,

the Geometric Center of the West District falls within the East District. A real-world application later on

in this paper includes comments on some specific potential consequences of such an inaccuracy. While

Geometric Centers are an adequate representative point in most instances, the presence of severely concave

regions can make them an ultimately inaccurate measure, and therefore unsuitable for use. Later in the

paper we will explore this statement further. In response to this, we will discuss a different point which

stays internal regardless of region shape. This work focuses on the defense of the validity of the proposed

point (which we refer to as the Concave Interior Center), a recommended associated inter-regional distance

measurement, and their applications to continuous facility location problems.

In the vast majority of real-life applications, it will be very difficult to describe all observed regions as

non-concave. Though an attempt can be made to cut a map into convex polygons, any application dealing

with infeasible sub-regions may run the risk of including concave regions. In fact, any instance of a

sub-region existing not against the border of its relevant region will create concavity. The noted issue of

using an externally located center is therefore one of under-discussed ubiquity. The potential severity of

such errors should not be understated. Any application with sufficient sensitivity is at risk of being greatly

affected.

The negative results of decisions made based on externally located regional centers fall into two main

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camps. Either an assumed solution is actually infeasible, or an assumed solution is feasible but inaccurate.

The former may come in the form of a recommendation to construct a warehouse in the middle of a lake,

the lake being where the Geometric Center of a certain distribution region happened to fall. This would

eventually be very obvious, and would force a manual location choice which would need to be universally

agreed upon, thus setting back the construction timeline. The latter is potentially more dangerous, as

it may become apparent later on and even after completion. For example, we are considering a project

which proposes to build a school located most conveniently to serve the majority of a population, and to

be outside of an interior "high crime" area located towards the center of the overall educational district.

Though it may not be apparent during the construction process, the long-term safety of children and school

faculty may be put at risk because of the unrealized fact that the Geometric Center of the district happened

to fall inside that intended exclusion zone. These will be specifically investigated again later with regards

to the application in section 5.

The remainder of this paper is separated into five primary sections. The coming section is a literature

review and discussion of existing center finding techniques, distance measurements, prior studies into

dealing with forbidden regions, and relevant applications, primarily in solving facility location placement

problems. Next, we describe our method for center identification and exemplify its use in problem solving.

Following that is an analysis of three specific regional occurrences which show notable properties of our

proposed Concave Interior Center. Afterwards we study the performance of our center in a real-world

continuous facility location application which focuses on maximizing services to a homeless population, a

currently critical humanitarian issue. Finally we conclude the paper, summarizing our observations and

results and noting the intended impact of the work and its potential for further exploration. In this paper

we will show our Concave Interior Center to be not only valid, but explicitly necessary in solving facility

location problems with concave regions in cases where other methods would underperform or fail.

2 Literature ReviewOur research intends to contribute to the study of center finding methods in the presence of forbidden

regions. The findings and results of such pursuits are most often utilized in distance measuring algorithms

and within the scope of facility location problems, but can be found in other applications such as stability

analysis for electrical systems as seen in the work of Liao et al. (2017) and Zhou et al. (2019)8, 9. Referring

to the problem of representing the regions, point specifying and sub-area designating methods have both

been used. Murray (2003) discussed the uncertainty and error in realistic facility locations compared to

initially suggested ones based on land suitability, access, and costs1. Murray & Kelly (2002) developed an

approach for evaluating representational appropriateness of the geographic space2. These papers opened

discussion on the problem on which our work focuses. Katz & Cooper (1979) brought about one of the

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earliest mentions of forbidden regions in location analysis3. Their focus looked at the relatively simple

case of placement in the case of a single circular forbidden region. Much later, we see similar work on

the case of two elliptical regions by Prakash et al. (2017)4. The same team expanded to the inclusion of

general polygonal and elliptical forbidden regions in Prakas et al. (2018)5. The sparsity of work in this

scope is clear in the time it took for more complex study to arise. Notable results on the computational

complexity of multifacility problems in this context have only recently been studied, as seen in Maier &

Hamacher (2019)10.

Wei et al. (2006) tried to solve the p-center problem in continuous space. They looked at the case that

a region is non-convex and possibly containing holes, thus causing an infeasible center. They proposed

generating a constrained minimum covering circle and using its center as the new location for calculating

distance to other centers6. Dinler et al. (2015) also used this point to represent regions and mentioned

techniques for studying to minimum and maximum region-facility distances, as well as expected distances.

They used the maximum distance to represent the region to calculate the distance of a demand region

to a facility location point and all the demand regions are closed and convex, which is a rectangle11.

Love (1972) considered the weighted sum of density and frequency of demand in an area to represent

its center, and exclusively represented regions and their infeasible sub areas as rectangles as an attempt

avoid non-convexity. Instead of using points to represent the region, Wei & Murray (2015) discussed

using collections of polygons to represent demand regions. Unlike a single point, which only assumes no

coverage or all coverage, polygonal representation takes into account percentage coverings when deciding

the facility locations12. Similarly, Murray & Tong (2007) depicted the region by square polygons to avoid

non-convexity. They used the intersect points of covering boundary of the polygon to be the potential

centers for the facility in the covering problem13.

P-norm distance, or "lp" distance, is widely used in the facility location problem to calculate distance

between two points. The following equation for "lp distance" was proposed by Francis (1967)14.

lp(q,r) =

[n

∑i=1|qi− ri|p

] 1p

= ||q− r||p = ||−→rq||p (1)

In equation 1, q = (q1, . . . ,qn) and r = (r1, . . . ,rn) are two points in n-dimensional space (i.e. q,r ∈ Rn).

Love & Morris (1972) compared seven different distance functions including lp functions, weighted lp

functions, and weighted Euclidean functions on two-road distance samples to determine their accuracies15.

They considered two criteria with respect to the accuracy of estimation:

• Sum of absolute deviations (the difference between actual road miles and estimated road miles).

• Sum of squares (more sensitive and require more accuracy on shorter distance).

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Love & Morris (1979) further compared different parameters for the “lp distance” model to make the

distance calculation more accurate16. They used statistical tests to compare the results on seven samples

and concluded that “the more parameters used in the model, the more accurate and complexity will get”.

Love Dowling (1985) used an “lk,p distance” model for floor layout problems to find the appropriate k, p

values compared to the “l1 distance” (or rectangular distance) model17. In the literature, authors stated that

the “l1 distance” model was suitable for obtaining optimal facility locations while “lk,p distance” model

should be used when total cost function is considered by adjusting k values rather than p values. None

of the several mentioned generalizations of this algorithm, unfortunately, can successfully determine the

exact location of the potential point in each region.

Batta et al. (1989) solved the location problem with convex forbidden regions using the Manhattan

metric (p = 1)18. Amiri-Aref et al. (2011) used a P-norm distance model to find the location for a new

facility which has maximum distance to other existing facilities with respect to the same weights and

line barriers19. They emphasized the rectilinear distance metric (p = 1) in the formulation and stated that

further research can be extended on different distance functions. Amiri-Aref et al. (2013) also presented

the same distance metric (p = 1) to solve the Weber location problem with respect to a probabilistic

polyhedral barrier20. They transferred the barrier zone from a central point of a barrier with extreme points

to a probabilistic rectangular-shaped barrier. Later, Amiri-Aref et al. (2015) utilized the “l1 distance”

model to deal with relocation problems with a probabilistic barrier21. Three approaches for solving

the restricted location-relocation problem with finite time horizon have been well-established in prior

literature. Dearing Segars (2002) presented that for solving single facility location problems with barriers

using rectilinear distance model (p = 1), the result remains the same for original barriers and modified

barriers22. Aneja & Parlar (1994) discussed two algorithms (Convex FR and Location with BTT) that

can solve Weber Location problems with forbidden regions for 1 < p ≤ 2 when using the lp distance

model to calculate the distance between demand points and facility location points23. Bischoff & Klamrot

(2007) illustrated a solution method for locating a new facility while given a set of existing facilities in

the presence of convex polyhedral forbidden areas24. This method first decomposes the problem, then

uses a genetic algorithm to find appropriate intermediate points. In existing literature, only Euclidean

distance (p = 2) has been considered, and it has been shown that this algorithm is very efficient. Altınel

et al. (2009) considered the heuristic approach for the multi-facility location problem with probabilistic

customer locations25. To calculate the expected distance between probabilistic demand points and facility

locations, they used Euclidean distance, squared Euclidean distance, rectilinear distance, and weighted

l1,2-norm distance models.

By way of network theory, we can find another way of calculating distances for location problem.

Blanquero et al. (2016) represented the competitive location problem as a network with customers as the

nodes26. They put the facilities on the edges and calculated the distances as the lengths of the shortest

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path from each customer to the nearest facility. Peeters (1998) discussed a new algorithm to solve the

location problem by calculating distance as the sum of the lengths of arcs between two vertices and the

length between a vertex and an arbitrary point on that arc27. The author used nine different models to

test the new algorithms, and showed much better performances than the old algorithm that calculates

the distance between each node first. One of the earlier uses of graphs in the context of the p-median

problem was in the work of Teitz & Bart (1968)28. Though their method of single vertex substitutions

was relatively inefficient, it was of significant importance for expansions on network based solutions

for these problems. Eilon & Galvão (1978) looked at double vertex substitutions in the uncapacitated

case of location analysis29. Jacobsen (1983) later used several heuristics and explored the context of the

capacitated problem30.

The intent of this paper is to further contribute to the study of continuous consideration of problems

on concave polyhedral maps, issues surrounding concave regions have been somewhat overlooked.

Satyanarayana et al. (2004) explored using an iterative line search method to find near-optimal placements,

which was extendable to non-convex settings31. Another iterative approach for near-optimal solutions in

this context was in the big triangle small triangle method proposed in Drezner & Suzuki (2004), which

itself was an extension of the generalized big square small square method of Plastria (1992)32, 33.

The work that has gone into solving optimal placements in concave settings has been restricted to iterative

approximating methods such as those described above, and advancements have been relatively sparse.

To the best of our knowledge, there is no current method which focuses on extracting points which most

accurately represent concave regions. Our view is that if a point is used to describe a region, but gives

unrealistic or unusable results in a model’s output, that point is an inaccurate. We assert the nature of the

application to be novel. Facility location research is typically tested on cost-minimizing full covering

models, most commonly with uncapacitated facilities. The application proposed in this paper concerns

capacitated facilities in the setting of an objective function to maximize reward under a specified budget.

The goal of this paper is to help fill these existing gaps, to expand the scope of applications on which facility

location algorithms may be tested, and to draw attention to the needs of existing homeless populations.

3 Concave Interior CenterWe now mathematically define our proposed Concave interior center and distance measuring met-

rics. The distance between two points in two dimensional space (x0,y0) and (x1,y1) can be defined

as√

(x0− x1)2 +(y0− y1)2. The point which minimizes the potential distance between itself and any-

where else in the region can be viewed functionally as a center. In light of this, we can use the aggregation

of distances between some (x∗,y∗) and the rest of the points the surrounding region R as a metric of

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centrality.

Suppose we have a region Si. We intend to use a point (xSi,ySi) ∈ Si, such that the value of∫ ∫(x,y)∈Si

√(xSi− x)2 +(ySi− y)2dxdy (2)

will be minimized. The distance between any two regions S1 and S2 with Concave Interior Centers

(xS1,yS1) and (xS2,yS2) is approximated as

d(S1,S2) =√

(xS1− xS2)2 +(yS1− yS2)

2 (3)

We see a discrete example of this in Figure 2 below. The black squares below all represent the same given

region. We look at the performances of three of its points. This is done by finding the sum of the distances

from the tested point (marked in red) to all other points within the region. Unchecked edge points will

yield identical results to what those shown in the middle and rightmost boxes respectively. Since the

Concave Interior Center is defined as the point minimizing aggregate distance, it is chosen as the one

highlighted in the leftmost box.

Figure 2. Example points. From left to right they have aggregate distances of 9.7, 12.3, and 14.7 units.

In most applications, the point (xSi,ySi) will be found in a discrete context. The density of points to

chose from in region Si, then, is fully up to the discretion of the operator. When utilized in this paper, the

number of available points is set as a bijection with the pixels in the displayed maps. For more precise

results, the map could have just as well been cut into an arbitrarily finer mesh. We assert that the center

point (xSi,ySi) we defined above well represents region i.

Our general argument that the point well represents the region is that by construction (xSi,ySi) is the point

which minimizes aggregate travel distance over the regional space.

4 Special Case AnalysisIn this section, we present three special cases to showcase notable qualities of our proposed point. In each,

we present a particular set of regions, and compare the resulting Concave Interior Centers and Geometric

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Centers. We find immediate and concrete differences in the performances. A more detailed discussion of

how these differences reflect our center’s importance can be found in following section.

Case 1A frequent occurrence faced in location placements is the existence of annular or annular-like concave

regions. We exemplify this in the following images. In Figure 3, we see that when using Geometric Center,

Regions I and II will act as having zero distance from each other, when clearly this is not the case. From

Figure 4, we can avoid this inaccuracy by using our Concave Interior Centers as representative points. We

can see from the generated maps that the new point gives a much more accurate distance inter-regional

distance.

Figure 3. Geometric Centers for case 1.

Figure 4. Concave Interior Centers for case 1.

Case 2We can see from Figure 5 and Figure 6 that this is an interesting case where the Geometric Center 1,

Geometric Center 2, Concave Interior Center 1, and Concave Interior Center 2 all fall identically. This

is a very specific case which shows the one weakness of our algorithm: that the algorithm is not able

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to identify the boundary of a region as potentially invalid, and therefore two separate regions may be

seen as having a distance of zero. A practical solution to this potential error is to make the boundaries of

all regions itself a region with any nonzero width, as any regions resulting in this situation will exist in

extremely close proximity. Though the resulting distance may be inaccurate, it will likely function in a

truthful manner. It is important to recognize that boundary positioned Concave Interior Centers will rarely

cause a critical error, as was seen in the annular-like case.

Figure 5. Geometric Centers for case 2.

Figure 6. Concave Interior Centers for case 2.

Case 3In this case, we have 3 regions. We present this case in the context that region 2 has a demand needing

to be served by either region 1 or 3. From Figure 7, we can see that the distance between the Geometric

Centers generated for regions 1 and 2 is smaller than that for regions 2 and 3, which implies that region 1

would be chosen as the servicer. It is shown in Figure 8 that under Concave Interior Centers the distance

between regions 2 and 3 is seen as smaller than the distance between regions 1 and 2. The demand at

region 2 would then be served by region 1. The latter can be visually confirmed as the better interpretation,

as the shortest distance between one point in each of regions 2 and 3 is smaller than that between regions

2 and 1 aside from the Concave Interior Center of region 1. The only case where this is not true is when

considering the region 2 point to be on a short excerpt of the theoretically single point wide horizontal line

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extending from where regions 1 and 2 actually touch (which is the marked Concave Interior Center of

region 1) up until approximately one-third of the way into region 2.

Figure 7. Geometric Centers for case 3.

Figure 8. Concave Interior Centers for case 3.

5 Real World ModelIn this section, we describe an important societal location problem to show the importance of the proposed

center. The model we have developed to solve this problem provides an opportunity to show the explicit

advantages of using the Concave Interior Center.

5.1 Problem DescriptionThe goal of this model is to find optimal placement of fixed capacity night-by-night homeless shelters and

decide whether or not to allocate funding for running food pantries at one or more of them in order to best

serve the needing populations of a given set of regions under offered funding.

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Any shelters constructed will have a universally set capacity. Funds received will be considered as a

one-time lump sum. The cost of shelter construction in a region will be approximated based on that of

comparable local buildings. Pantry cost is determined as the price of one daily meal for each homeless

resident of its constructed region, funded for the span of ten years. Regional borders for the map will

be drawn in response to the following phenomena: existing town boundaries, areas unavailable for

construction due to natural features, and areas not zoned for construction of such shelters.

The region observed is Fairfield County, Connecticut. This county well shows the importance of accurate

regional representation for a number of reasons. The difference in approximated costs, described below,

between even adjacent towns is seen notably between Stamford ($339,000) and Darien ($1,595,000). The

map features many lakes, mountainous areas, wetlands, and protected forests in which construction is

impossible. Zoning regulations, additionally, vary greatly. Greenwich ordinances, for example, flat out

ban such facilities, while Danbury is largely open for construction. Ridgefield, however, does not currently

have defined legislation for this form of shelter, as is often the case. When information is not available

it will be assumed that exclusively residential, full industrial, agricultural, historic, and aesthetic zoning

districts will be marked as infeasible, as these areas are the most tightly restricted in terms of building

type.

Set up for this application required researching four topics. Specifically, we needed to set up a map

to depict areas valid and not valid for construction, find homeless populations for each region, and

approximate build cost for each. The valid districts are set in the map as tan, invalid build districts are

marked red, and ignored areas of the map are marked grey (which will be treated the same as invalid

districts, but with no demand). Specific reasons for ineligibility are not noted for two reasons: there

is significant overlap between aforementioned causes (many lakes already lie in exclusive residential

zones), and the only practical difference between causes is whether or not there could still be a homeless

population in the area. Homeless populations are approximated based on data from Point-in-Time counts

by the Connecticut Coalition to End Homelessness, and split into regions within towns based on their

relative population densities34. Facility cost was found for each town by increasing or decreasing the

approximated construction cost for Danbury based on the relative percent increase or decrease housing

costs (land and construction). Danbury’s cost was an approximation based on that of its own homeless

shelter, the Dorothy Day Hospitality House. Data for pantry costs is approximated by statistics from

Feeding America and the Regional Food Bank of Northeastern New York35–37. When information is not

available or has not yet been legislated on for zoning validity, as is the case with Ridgefield, it will be

assumed that any area marked as exclusive residential, full industrial, agricultural, historic, and aesthetic

type zones will be marked as infeasible, as these areas are the most tightly restricted in terms of building

type. For our initial application, shelters will universally have a 45 bed capacity, and the budget for the

overall project will be $2,000,000. Figure 9 below is the section of southwestern Connecticut we are

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referring to. A table containing relevant attributes of each region is shown in tables 3,4, and 5 following

the discussion in section 5.3

Figure 9. Left: Fairfield County split into valid (tan) and invalid (red) regions. Right: A map of the

county split on town borders.

5.2 Model

We now define the model we will use to solve our decision problem. Tables 1 and 2 list the associated

variables and parameters along with their definitions in the context of the problem. The objective function

and constraints are presented in equations (4) through (16).

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Table 1. Definitions for model parameters.

parameters definition

ci cost of constructing a shelter at center of region i

Di Demand (homeless population) of region i

fi fi = 1 if construction of a shelter at region center i is

feasible, else fi = 0

R Set of indices of Region centers for r regions

{1,2, ...,r}L Set capacity of all shelters

v Weighting of value added by inclusion of a pantry

p Cost of running a pantry

B Overall budget for construction of shelters

M Large cost parameter

Table 2. Definitions for model variables.

variables definition

ei ei = 1 if a shelter is to be constructed at region center

i, else ei = 0

Fi Fi = 1 if a pantry is to be operated through the shelter

at region i, else Fi = 0

sm,n sm,n = 1 if demand of region m is satisfied by facility

at region center n, else sm,n = 0

dm,n distance between center point of regions n and m

hi hi = 1 if Di ≥ L , else hi = 0

13

maxr

∑i=1

r

∑n=1

[Lsi,nhi +Disi,n(1−hi)]+ vr

∑i=1

[LFihi +DiFi(1−hi)] (4)

sm,n ≤ en ∀m,n ∈ R (5)

Fi ≤ ei ∀i ∈ R (6)

ei ≤ fi ∀i ∈ R (7)

sm,n ≤ Dm ∀m,n ∈ R (8)

dm,nsm,n ≤ dm,i +M(1− ei) ∀m,n, i ∈ R (9)r

∑m=1

r

∑n=1

sm,n =r

∑i=1

ei (10)

r

∑i=1

(ciei +2100DiFi) ≤ B (11)

r

∑n=1

sm,n ≤ 1 ∀m ∈ R (12)

r

∑m=1

sm,n ≤ 1 ∀n ∈ R (13)

r

∑m=1

sm,n ≥ en ∀n ∈ R (14)

dm,nsm,n ≤1r2

r

∑i=1

r

∑j=1

di, j ∀m,n ∈ R (15)

dm,nem +dm,nen

2+M(1− em)+M(1− em) ≥

12r2

r

∑i=1

r

∑j=1

di, j ∀m,n ∈ R (16)

The goal of the objective function (4) is to maximize the homeless population served, automatically

setting the population served by any shelter to equal capacity L if the region being served by that shelter

has a demand greater or equal to said than said capacity. Feeding a person through the pantry is considered

a fraction (v) of the value of sheltering the person. Constraint (5) ensures that only existing shelters can

serve a region. Constraint (6) requires that pantries can only be created at shelter sites. Constraint (7)

stops shelters from being constructed in feasible regions. Because it is unrealistic that a person would

travel an unnecessary additional distance, constraint (9) states that a region can only be served by its

nearest shelter. If an optimal solution is found with sufficient room left within the budget, the model may

request the construction of a non servicing shelter. Constraints (10) and (14) prevents this. If, in addition,

there is a low cost feasible construction zone near on or near a region with no demand, the model may

request construction of a shelter which does not contribute any objective improvement. This is inhibited

by constraint (8). Per constraint (11), the sum of the construction and food pantry costs must not exceed

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the available budget. Constraint (12) limits a region’s population to be served by at most one shelter.

Otherwise, any constructed shelter would be automatically assumed to have demand exceeding capacity.

For the same reason, constraint (13) requires that any shelter serves at most one region. Constraints (15)

and (16) bound the distance between a shelter and the region it services as less than a fraction of the

aggregate of all inter-regional distances, and distance between any two shelters as greater than that same

fraction.

5.3 Computational Result AnalysisThe proposed algorithm for center calculation is implemented with MATLAB R 2019a on a computer with

Intel(R) Xeon(R) E3-1231 v3 CPU running at 3.4 GHz with 16 GB memory and a 64-bit operating system.

We get the Geometric Center for a region in practice by running the function (∑

nk=1 xk

n ,∑

nk=1 yk

n ) for the set of

n points in a given region, with the ith point represented as (xi,yi). The Concave Interior Center of a region

is found as the point (xSi,ySi) which minimizes the output of the function ∑nb ∑

ma√

(xSi− xa)2 +(ySi− yb)2

over the finite set (xa,yb) ∈ Si. The model is run on the regioned map in Figure 9 scaled to 638 x 474 pixels,

with pixels catalogued as matrix form (i.e. origin at top left as (1,1) and address as (row, column)). The

model is implemented with Julia Language 0.6 on a Macbook pro with 3.5 GHz Intel Core I-7 processor

and 16 GB memory with Julia Language 0.6 and solved with Gurobi 7.5.2 solver.

Tables 3,4, and 5 show each region’s homeless population, cost of construction, construction feasibility,

Geometric Center, Concave Interior Center, and respective town. For use in the model constraints, con-

struction cost in infeasible regions is set as 0. Tables 6 and 7 show the results of running the model. The

Parameters column show the parameter sets v,L, and B for each run. The Geometric column shows the

objective value, the regions at which shelters should be constructed n, the region which each constructed

shelter should serve m, and whether or not a food pantry should be run at the shelter, assuming the model

uses the Geometric Centers from tables 3,4, and 5. The Concave column shows the same information

when using the Concave Interior Centers.

A picture by picture inspection shows that all discrepancies between region centers occur in cases where

the Geometric Center falls external to that region. The listed centers in tables 3,4, and 5 for regions 1, 2, 5,

6, 16, 22, 30, 45, 46, 47, and 48 are highlighted to note their differences. It is immediately apparent that

under all parameter sets, regions with populations which met or exceeded shelter assigned shelter capacity

were favored. This is most evident in regions 13, 46, 47, and 61 as being most commonly chosen for

parameter sets when L = 100. When shelter capacities are reduced to 45, it is notable that highest priority

is placed on low cost construction regions. This can be attributed to having a larger pool of capacity

fulfilling choices. Under this same capacity with B = 4,000,000 and v = 0.5 using Geometric Centers,

it is notable that fewer than half of the regions serviced have less demand than the shelter capacity, and

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that 6 of the shelters include food pantries. When changing to v = 0.125, we see that more than half of

the serviced regions have demand greater than capacity, and that only 4 of the 7 shelters have pantries. It

is evident that this occurs because less priority is placed on the benefit which results from the objective

addition of the chosen construction regions’ population being served through associated food pantries.

Had the same solution set been for v = 0.125 as was chosen for v = 0.5, the objective value would have

fallen to 274.375, as opposed from the achieved 282.5. This difference is illustrated in Figures 10 and 11.

The differences when using our concave interior center when compared to the Geometric Center are of

critical importance. In each run with differing objective values, we see that one or more of the regions

chosen under the Geometric Center have construction locations which are different from their concave

interior centers and which, as previously noted, are actually infeasible. It is notable that this results in a

higher objective value which, in practice, would be impossible to obtain. We can look at the implications

of this in practice through our model. Under the parameters v = 0.5,L = 45,B = 4,000,000, construction

for regions 45, 47, and 48 would be infeasible. Removing these would drop the resulting objective from

329.5 to only 174.5. We note that the objective will not always be improved. Looking to Special Case 1

we can imagine a third, much smaller concentric region creating a third innermost region. If we wanted

to maximize distance between regions while choosing only two regions, Geometric Center would imply

any pair would give identical results. concave interior centers, however, would imply that the two regions

which are furthest apart radii, the inner most and outermost ones, would give the maximum distance.

It is easy to imagine a scenario in which the executor of this project is chosen based on projected perfor-

mance. In such an event, a less considerate party which assumes efficacy of the Geometric Center may be

granted the contract. The results would be disastrous, as construction may be started before realizing the

impending error. Following that, the board in charge of the approved grant would need to make one of

two choices. They could either nullify the contract and give it to the more considerate firm which uses

Concave Interior Centers, now with a much lower budget, or allow the initial company to only commit to

the construction of their feasible shelter choices, resulting in significantly reduced objective value. Our

model explicitly shows the importance of our proposed point in avoiding the critical error that can arise

from a naïve choice of estimator.

16

Table 3. Data and calculated centers for regions 1-21.

Region Di Ci Fi Geometric Concave Town

1 20 0 0 (85,167) (85,166) Sherman

2 0 0 0 (113,178) (112,179) Sherman

3 15 480000 1 (115,171) (115,171) Sherman

4 0 384000 1 (144,198) (144,198) New Fairfield

5 0 0 0 (164,192) (163,192) New Fairfield

6 6 384000 1 (151,156) (150,156) New Fairfield

7 42 0 0 (183,166) (183,166) New Fairfield

8 23 384000 1 (173,172) (173,172) New Fairfield

9 32 394000 1 (179,220) (179,220) Brookfield

10 27 0 0 (196,231) (196,231) Brookfield

11 25 394000 1 (192,251) (192,251) Brookfield

12 0 0 0 (212,198) (212,198) Danbury

13 123 320000 1 (247,184) (247,184) Danbury

14 0 0 0 (255,183) (255,183) Danbury

15 20 357000 1 (267,223) (267,223) Bethel

16 5 0 0 (253,307) (252,307) Newtown

17 40 0 0 (277,245) (277,245) Bethel

18 2 428000 1 (250,317) (250,317) Newtown

19 10 686000 1 (283,160) (283,160) Ridgefield

20 0 0 0 (309,191) (309,191) Danbury

21 10 0 0 (339,159) (339,159) Ridgefield

17

Table 4. Data and calculated centers for regions 22-42.

Region Di Ci Fi Geometric Concave Town

22 20 686000 1 (346,178) (346,179) Ridgefield

23 9 627000 1 (367,209) (367,209) Redding

24 11 0 0 (336,243) (336,243) Redding

25 0 707000 1 (334,297) (334,297) Easton

26 4 0 0 (311,357) (311,357) Monroe

27 2 418000 1 (302,340) (302,340) Monroe

28 6 418000 1 (326,349) (326,349) Monroe

29 32 0 0 (337,411) (337,411) Shelton

30 24 364000 1 (351,438) (351,435) Shelton

31 80 0 0 (409,419) (409,419) Stratford

32 50 0 0 (380,370) (380,370) Trumbull

33 35 424000 1 (383,343) (383,343) Trumbull

34 7 0 0 (383,342) (383,342) Trumbull

35 8 0 0 (382,309) (382,309) Easton

36 5 707000 1 (390,319) (390,319) Easton

37 0 707000 1 (400,297) (400,297) Easton

38 2 707000 1 (378,274) (378,274) Easton

39 0 0 0 (396,253) -396,253 Weston

40 0 873000 1 (403,259) (403,259) Weston

41 0 873000 1 (418,242) (418,242) Weston

42 11 0 0 (427,206) (427,206) Wilton

18

Table 5. Data and calculated centers for regions 43-63.

Region Di Ci Fi Geometric Concave Town

43 0 873000 1 (426,275) (426,275) Weston

44 19 0 0 (454,319) (454,319) Fairfield

45 21 617000 1 (465,345) (466,346) Fairfield

46 162 0 0 (441,372) (443,374) Bridgeport

47 150 192000 1 (444,378) (443,375) Bridgeport

48 42 278400 1 (449,414) (448,414) Stratford

49 48 0 0 (465,415) (465,415) Stratford

50 7 1288000 1 (482,271) (482,271) Westport

51 0 0 0 (508,281) (508,281) Westport

52 0 0 0 (488,263) (488,263) Westport

53 94 0 0 (505,217) (505,217) Norwalk

54 45 466000 1 (509,229) (509,229) Norwalk

55 0 0 0 (534,227) (534,227) Norwalk

56 65 466000 1 (523,212) (523,212) Norwalk

57 2 1508000 1 (450,158) (450,158) New Canaan

58 3 0 0 (483,171) (483,171) New Canaan

59 6 0 0 (542,178) (542,178) Darien

60 0 1595000 1 (543,163) (543,163) Darien

61 180 0 0 (514,122) (514,122) Stamford

62 40 339000 1 (563,136) (563,136) Stamford

63 35 0 0 (549,67) (549,67) Greenwich

19

Table 6. Changes in chosen serviced regions m and respective servicing shelter construction regions n

under Geometric and Concave Interior Centers for differing shelter capacities L and Budgets B with

v = 0.5.

Parameters Geometric Results Concave Resultsv L B Objective sm,n Pantry Objective sm,n Pantry

0.5 100 2,000,000 428.5 13,6 yes 428.5 13,6 yes

32,18 yes 32,18 yes

46,23 yes 46,23 yes

53,62 yes 53,62 yes

0.5 100 4,000,000 547 10,3 yes 547 10,3 yes

13,22 yes 13,22 yes

46,38 yes 46,38 yes

47,28 yes 47,28 yes

53,40 no 53,40 no

61,23 yes 61,23 yes

0.5 45 2,000,000 232.5 13,6 yes 223.5 12,3 yes

17,4 no 17,27 yes

29,47 no 49,33 yes

30,48 yes 61,54 yes

61,54 yes

0.5 45 4,000,000 329.5 7,19 yes 300.5 13,6 yes

8,15 yes 15,4 no

29,47 yes 17,28 yes

30,48 yes 22,41 no

53,37 no 46,43 no

54,45 yes 49,30 yes

61,23 yes 61,54 yes

20

Table 7. Changes in chosen serviced regions m and respective servicing shelter construction regions n

under Geometric and Concave Interior Centers for differing shelter capacities L and Budgets B with

v = 0.125.

Parameters Geometric Results Concave Resultsv L B Objective sm,n Pantry Objective sm,n Pantry

0.125 100 2,000,000 407.125 13,6 yes 407.125 13,6 yes

32,18 yes 32,18 yes

47,23 yes 46,23 yes

53,62 yes 53,62 yes

0.125 100 4,000,000 530.875 9,27 yes 530.875 9,27 yes

13,22 yes 13,22 yes

46,38 yes 46,38 yes

47,28 yes 47,28 yes

53,40 no 53,40 yes

61,23 yes 61,23 yes

0.125 45 2,000,000 273.5 13,6 yes 273.5 13,6 yes

31,28 yes 31,28 yes

49,37 no 49,37 no

53,62 yes 53,62 yes

0.125 45 4,000,000 282.5 9,18 no 273.25 13,6 yes

13,22 yes 17,27 yes

29,47 yes 31,28 yes

30,48 yes 46,43 no

53,37 no 49,50 yes

54,45 no 61,54 yes

61,23 yes

21

Figure 10. Results from v = 0.125;L = 45;B = 4,000,000 and Geometric Centers (left) and Concave

Interior Center (right). Shelters with food pantries are marked green, Shelters without are marked blue.

Figure 11. Results from v = 0.5;L = 45;B = 4,000,000 and Geometric Centers (left) and Concave

Interior Center (right). Shelters with food pantries are marked green, Shelters without are marked blue.

22

6 ConclusionCenter finding methods are of critical importance in location analysis and related humanitarian and

scientific applications. Despite the amount of research existing on dealing with forbidden regions in such

problems, there has been a distinct lack of focus on the issues arising from nonconvex contexts. In this

work, we presented a novel method for finding representative regional center points, which we referred

to as Concave Interior Centers, by solving a minimization function over the chosen mesh of possible

locations. We created a real-world model to test the validity of our generated points against that of the set

of Geometric Centers. Three special cases were analyzed to show notable characteristics of our proposed

center. These included a commonly occurring instance of severe inaccuracy of the Geometric Center, the

possible inaccuracy of border points being chosen as Concave Interior Centers, and a less severe spacial

inaccuracy which still causes a significantly different conclusion for an optimized solution. We showed

this in an application which had the goal of maximizing the amount of social services offered to homeless

people sheltered in the context of limited capacity shelters and a limiting shelter construction budget.

In practice, we would hope to have more guaranteed data for our model. The point-in-time counts used for

our model were not given with respect to each region described on our map, but were rather grouped by one

or more towns. The splitting of populations had to be done by estimation based on nature and approximate

population density of each region and its respective town. Furthermore, the approximation of shelter costs

could be greatly impacted by potentially preexisting sites which could be renovated for use. Regardless,

our model results offered explicit proof that, in practice, our centers avoided the unrealistic and flawed

solutions which were offered by more naïve methods. Further study could be made into the computational

complexity for finding the Concave Interior Center. Going forward it may also be worthwhile to look

into its mathematical relationship to other methods, specifically to see what circumstances would cause

identical points to be generated by, say, the centers of regionally inscribing or circumscribing circles.

The center finding method proposed in this paper has been shown to be sound, effective, and advantageous

in use. We put forth that it is not only appropriate for use in any application which hinges on finding

regionally representative points. Moreover we argue that it is the most appropriate existing method for

finding such points in any context which desires to minimize travel distance. This extends to any number

of applications from supply chains to disaster relief efforts.

23

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